Automorphism groups of real rational quartic del Pezzo surfaces

Abstract: In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in $\mathbb{P}<sup>{3}_{\mathbb{R}}$. We examine all possible ways to blow up $4$ geometric points on $Q$, illustrate in each case the $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-action on the conic bundle structures on $X_{\mathbb{C}}$, and use it to give a geometric description of the real automorphism group $\operatorname{Aut}<em>{\mathbb{R}}(X)$ by generators in terms of automorphisms and birational automorphisms of $Q$. As a consequence, we get which finite subgroups of $$\operatorname{Bir}</em>{\mathbb{C}}(\mathbb{P}<sup>{2})$$ can act faithfully by automorphisms on real $\mathbb{R}$-rational del Pezzo surfaces of degree $4$.

• The paper classifies all five types of real rational quartic del Pezzo surfaces by expressing them as blow-ups of two real quadrics and computing their automorphism groups through actions on five exceptional pairs.
• The classification gives kernels ranging from $(\mathbb{Z}/2)^2$ to $(\mathbb{Z}/2)^4$, while the possible image groups include trivial, involutive, symmetric, and 5-cycle-containing subgroups determined by precise parameter conditions.
• The results show that a surface’s Galois action on its Picard group is generally not realized by biregular automorphisms, with exceptional symmetries arising only at special arithmetic values of the moduli parameters.

Overview and main result

This paper by Aurore Boitrel classifies the automorphism groups of all real rational del Pezzo surfaces of degree 4, i.e., smooth projective surfaces with ample anticanonical class satisfying $K_X^2 = 4$ that are rational over $\mathbb{R}$. The classification is achieved by realizing each such surface $X$ as the blow-up of one of the two non-isomorphic smooth real quadric surfaces in $\mathbb{P}^3_{\mathbb{R}}$ — namely $Q_{2,2}$ (isomorphic to $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$) or its non-split form $Q_{3,1}$ — at four geometric points, and then describing $\Aut_{\mathbb{R}}(X)$ via the Galois action on conic bundle structures. The work completes and extends prior partial results of Robles [rob16] and Yasinsky [yas22].

The central technical device is the homomorphism $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$ induced by the action on the five exceptional pairs of divisors (equivalently, the five pairs of conic bundle structures), whose kernel embeds into $(\mathbb{Z}/2)^4$. Restricting to the real automorphism group yields an exact sequence

$\mathbb{R}$0

where $\mathbb{R}$1 and $\mathbb{R}$2. The main theorem states that every real rational quartic del Pezzo surface is one of five types, summarized below.

Surface	Blown-up points	Parameter	$\mathbb{R}$3	Possible $\mathbb{R}$4
$\mathbb{R}$5	4 real points	$\mathbb{R}$6	$\mathbb{R}$7	$\mathbb{R}$8, $\mathbb{R}$9, or trivial
$X$0	2 real + 1 conjugate pair	$X$1	$X$2	$X$3 or trivial
$X$4	2 conjugate pairs	$X$5	$X$6	$X$7 or trivial
$X$8	4 real points	$X$9	$\mathbb{P}^3_{\mathbb{R}}$0	$\mathbb{P}^3_{\mathbb{R}}$1, $\mathbb{P}^3_{\mathbb{R}}$2, or trivial
$\mathbb{P}^3_{\mathbb{R}}$3	2 conjugate pairs	$\mathbb{P}^3_{\mathbb{R}}$4	$\mathbb{P}^3_{\mathbb{R}}$5	$\mathbb{P}^3_{\mathbb{R}}$6 or trivial

The paper also determines which finite subgroups of $\mathbb{P}^3_{\mathbb{R}}$7 can act faithfully by automorphisms on these surfaces.

Method: Galois action on conic bundles

The strategy proceeds case by case. For each real form $\mathbb{P}^3_{\mathbb{R}}$8, the author enumerates the sixteen $\mathbb{P}^3_{\mathbb{R}}$9-curves on $Q_{2,2}$0, collects the ten conic bundle structures into five exceptional pairs $Q_{2,2}$1 with $Q_{2,2}$2, and draws the action of the antiholomorphic involution $Q_{2,2}$3 on these pairs. This immediately constrains both $Q_{2,2}$4 (via conditions like $Q_{2,2}$5) and $Q_{2,2}$6 (as a subgroup of a dihedral group). The existence of specific automorphisms is then established by exhibiting explicit generators: involutions of $Q_{2,2}$7 of the form $Q_{2,2}$8 with $Q_{2,2}$9, together with birational involutions of bidegree $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$0 whose base points are precisely the blown-up points; these lift to biregular automorphisms of $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$1. Non-existence arguments combine Picard group computations (showing certain candidate actions yield matrices not in $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$2) with explicit coordinate calculations showing that required parameters would take prohibited values.

The case $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$3

For the blow-up of $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$4 at two conjugate pairs $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$5, normalized so that $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$6 and $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$7 with $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$8, the paper gives an alternative proof of results from [rob16]: $$\mathbb{P}^1_{\mathbb{R}} \times \mathbb{P}^1_{\mathbb{R}}$$9 is generated by elements realized as lifts of two real involutions and one birational involution of $Q_{3,1}$0, while $Q_{3,1}$1 if and only if $Q_{3,1}$2, and is trivial otherwise. Notably, for general $Q_{3,1}$3 (e.g., $Q_{3,1}$4), the Galois action on the Picard group is not realized by any automorphism of the surface — the antiholomorphic involution does not lift biregularly in general.

The case $Q_{3,1}$5

Here $Q_{3,1}$6 is the blow-up at two real points $Q_{3,1}$7 and one conjugate pair $Q_{3,1}$8, normalized as $Q_{3,1}$9, $\Aut_{\mathbb{R}}(X)$0, $\Aut_{\mathbb{R}}(X)$1. The normalization lemma shows that $\Aut_{\mathbb{R}}(X)$2 exactly when the blow-up fails to be del Pezzo. The main result for this type is that $\Aut_{\mathbb{R}}(X)$3, generated by the lift of an involution $\Aut_{\mathbb{R}}(X)$4 exchanging $\Aut_{\mathbb{R}}(X)$5 and $\Aut_{\mathbb{R}}(X)$6, and the lift of a birational involution $\Aut_{\mathbb{R}}(X)$7 with base points $\Aut_{\mathbb{R}}(X)$8. The image satisfies $\Aut_{\mathbb{R}}(X)$9 if and only if $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$0, and is trivial otherwise. The proof rules out actions of type $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$1, $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$2, and $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$3: the first two fail because candidate matrices lie outside $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$4 or force $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$5; the order-four element $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$6 would require a birational map of order 4 whose existence forces $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$7, which is excluded. Again, for general (non-real) $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$8, the Galois action is not realized by an automorphism.

The case $\rho_1 : \Aut(X_\mathbb{C}) \to \Sym_5$9

This is the richest case. The four blown-up points are normalized as $(\mathbb{Z}/2)^4$0, $(\mathbb{Z}/2)^4$1, $(\mathbb{Z}/2)^4$2, $(\mathbb{Z}/2)^4$3 with $(\mathbb{Z}/2)^4$4; the del Pezzo condition excludes $(\mathbb{Z}/2)^4$5. Here $(\mathbb{Z}/2)^4$6, generated by lifts of two involutions ($(\mathbb{Z}/2)^4$7 swapping $(\mathbb{Z}/2)^4$8, $(\mathbb{Z}/2)^4$9; $\mathbb{R}$00 swapping $\mathbb{R}$01, $\mathbb{R}$02) and one birational involution $\mathbb{R}$03. The image $\mathbb{R}$04 takes three possible values governed by sharp arithmetic conditions on $\mathbb{R}$05:

• $\mathbb{R}$06 if and only if $\mathbb{R}$07;
• $\mathbb{R}$08 if and only if $\mathbb{R}$09 and $\mathbb{R}$10;
• $\mathbb{R}$11 otherwise.

The element $\mathbb{R}$12 arises from an automorphism of $\mathbb{R}$13 of order three preserving both rulings and cyclically permuting $\mathbb{R}$14; its existence reduces to the quadratic relation $\mathbb{R}$15. The transpositions $\mathbb{R}$16 and $\mathbb{R}$17 individually never occur in the image, being excluded either by non-integrality of the induced Picard action or by forcing $\mathbb{R}$18, contradicting the hypothesis.

Remaining cases over $\mathbb{R}$19

The cases $\mathbb{R}$20 and $\mathbb{R}$21 are treated analogously using $\mathbb{R}$22. For $\mathbb{R}$23, with two real parameters $\mathbb{R}$24, the kernel is maximal, $\mathbb{R}$25, and the image can be as large as $\mathbb{R}$26, containing a 5-cycle — reflecting the fact that over $\mathbb{R}$27 the full symmetric structure on the five exceptional pairs is more accessible than over $\mathbb{R}$28. For $\mathbb{R}$29, with parameters $\mathbb{R}$30, the image is either $\mathbb{R}$31 or trivial, while the kernel remains $\mathbb{R}$32.

Scope and limitations

The classification is restricted to rational real forms of degree 4; non-rational real forms of quartic del Pezzo surfaces (those with real points but no real rational parametrization, or without real points) fall outside the scope of this paper, though the $\mathbb{R}$33 case treated here coincides with the rational locus of the family studied in [rob16]. The analysis relies on characteristic zero methods (the antiholomorphic involution, complex conjugation on parameters), so no statement extends directly to positive characteristic. The paper also leaves implicit the question of conjugacy classes of these finite groups inside $\mathbb{R}$34 beyond recording which subgroups occur faithfully as automorphism groups.

Conclusion

The paper provides a complete, generator-level description of $\mathbb{R}$35 for every real rational quartic del Pezzo surface, organized through the exact sequence $\mathbb{R}$36 attached to the five exceptional pairs. The kernels are elementary abelian 2-groups of rank 2 to 4, generated by explicit lifts of involutions and birational involutions of the underlying quadric, while the images are constrained dihedral subgroups of $\mathbb{R}$37 whose occurrence is governed by precise arithmetic conditions on one or two moduli parameters. A recurring structural finding is that for general parameter values the Galois action on the Picard group is not realized by any real automorphism, so the exact sequence genuinely fails to split over the Galois-fixed part in most of the moduli space.